Portfolio Volatility

[desh]

Can you help in solving the sum step by step
A portfolio is invested equally into two funds, each with normally distributed returns. The first fund has an expected return of 6.0% with return volatility of 8.0%. The second fund has an expected return of 10.0% with return volatility of 15.0%. The funds are independent (uncorrelated). Which is nearest to the probability that the portfolio return will exceed 12.0%?
ans is : The expected portfolio return is 8% = 50%*6% + 50%*10%. The portfolio volatility = SQRT(50%^2*8% + 50%*15%) = 8.50%. The Z value = (12% - 8%)/8.50% = 0.470588 such that NORM.S.DIST(0.470588, true = CDF) = 68.10% is Pr(R <= 12). Therefore, Prob (R < 12%) = 1 - Pr(R<=12%) = 1 - 68.10% ~= 31.90%

I am not getting how Portfolio volatility is calculated... Please guide.

 

[Deepak Chitnis]

Hi @desh , portfolio volatility is calculated as portfolio volatility =SQRT[weight of A^2*volatility of A^2+weight of B^2*volatility of B^2+2*weight of A*weight of B*volatility of A*volatility of B*correlation], as per question correlatio = 0 so, portfolio volatility =SQRT[weight of A^2*volatility of A^2+weight of B^2*volatility of B^2] now just plug in the values portfolio volatility = SQRT(50%^2*8%^2 + 50%*15%^2) = 8.50%. Hope that helps! and please try to post query in particular thread!
Thank you

 

[desh]

thanks @Deepak Chitnis Sir....Thanks for step wise clarification......

 

[Nicole Seaman]

Hello @desh

Just a note that I moved your thread from the Error thread to this correct section of the forum. The error threads are ONLY for members to point out mistakes that they find in our materials, more specifically, our study notes. There is also thread discussing this question in more detail in our paid section, but this section of the forum remains open to our free members so I moved your question here.

Thank you,

Nicole

 

[David Harper CFA FRM]

@desh @Deepak Chitnis Please note source is here @ https://forum.bionicturtle.com/threads/p1-t2-403-probabilities.7947
Please note the squared terms: portfolio volatility = SQRT(50%^2*8%^2 + 50%^2*15%^2) = 8.50%; i.e., if independence (--> ρ=0), the σ(P) = sqrt(w1^2*σ1^2 + w2^2*σ2^2)

 

[Deepak Chitnis]

Hi @David Harper CFA FRM fixed above, just realised that I forgot to take variance(used it while calculations, but forgot to write). You are great!
Thank you

 

[David Harper CFA FRM]

Thanks @Deepak Chitnis! I sometimes confuse myself on this seemingly basic formula, in the following way: because VaR(x+y) = SQRT[VaR(x)^2 + VaR(y)^2] when correlation is zero, in this case given that weights are equal, I instinctively first just assumed σ(x+y) = sqrt(8%^2 + 15%^2) = 17% because VaR acts just like volatility (as VaR is just a scalar multiple of volatility). But such a "shortcut" really needs the terms to be in dollars because dollars implicitly contain the weights. What I mean is that let's just assume by equal weight we have $100 invested into each fund.

• Then dollar volatilities are $8 and $15 such that ρ=0 implies $σ(x+y) = SQRT(8^2 + 15^2) = $17.00, which is correct
• But as the portfolio is $200, that's $17/$200 = 8.5% portfolio volatility. My "shortcut" works but only if you divide the dollar volatility back into the correct portfolio dollar, so it's probably just safer to directly use the correct SQRT(w1^2*σ1^2 + w2^2*σ2^2) = SQRT(50%^2*8%^2 + 50%^2*15%^2). In case that's interesting

 

[Deepak Chitnis]

Yes @David Harper CFA FRM it happens sometimes! As you said, I agree with using direct volatility formula, it is safer. And your shortcut is interesting! Thank you for sharing!

 

[desh]

Thanks @Deepak Chitnis! I sometimes confuse myself on this seemingly basic formula, in the following way: because VaR(x+y) = SQRT[VaR(x)^2 + VaR(y)^2] when correlation is zero, in this case given that weights are equal, I instinctively first just assumed σ(x+y) = sqrt(8%^2 + 15%^2) = 17% because VaR acts just like volatility (as VaR is just a scalar multiple of volatility). But such a "shortcut" really needs the terms to be in dollars because dollars implicitly contain the weights. What I mean is that let's just assume by equal weight we have $100 invested into each fund.

• Then dollar volatilities are $8 and $15 such that ρ=0 implies $σ(x+y) = SQRT(8^2 + 15^2) = $17.00, which is correct
• But as the portfolio is $200, that's $17/$200 = 8.5% portfolio volatility. My "shortcut" works but only if you divide the dollar volatility back into the correct portfolio dollar, so it's probably just safer to directly use the correct SQRT(w1^2*σ1^2 + w2^2*σ2^2) = SQRT(50%^2*8%^2 + 50%^2*15%^2). In case that's interesting

@David Harper CFA FRM your shortcut is amazing.....

 

[yLam4028]

Thanks @Deepak Chitnis! I sometimes confuse myself on this seemingly basic formula, in the following way: because VaR(x+y) = SQRT[VaR(x)^2 + VaR(y)^2] when correlation is zero, in this case given that weights are equal, I instinctively first just assumed σ(x+y) = sqrt(8%^2 + 15%^2) = 17% because VaR acts just like volatility (as VaR is just a scalar multiple of volatility). But such a "shortcut" really needs the terms to be in dollars because dollars implicitly contain the weights. What I mean is that let's just assume by equal weight we have $100 invested into each fund.

• Then dollar volatilities are $8 and $15 such that ρ=0 implies $σ(x+y) = SQRT(8^2 + 15^2) = $17.00, which is correct
• But as the portfolio is $200, that's $17/$200 = 8.5% portfolio volatility. My "shortcut" works but only if you divide the dollar volatility back into the correct portfolio dollar, so it's probably just safer to directly use the correct SQRT(w1^2*σ1^2 + w2^2*σ2^2) = SQRT(50%^2*8%^2 + 50%^2*15%^2). In case that's interesting

amazing.. to anyone seeing this this also works with portfolio with non zero correlation and more than 2 funds as long as they are equally funded.